A Taylor Series Approach to Correct Localization Errors in Robotic Field Mapping using Gaussian Processes

Gaussian Processes (GPs) are powerful non-parametric Bayesian models for regression of scalar fields, formulated under the assumption that measurement locations are perfectly known and the corresponding field measurements have Gaussian noise. However, many real-world scalar field mapping applications rely on sensor-equipped mobile robots to collect field measurements, where imperfect localization introduces state uncertainty. Such discrepancies between the estimated and true measurement locations degrade GP mean and covariance estimates. To address this challenge, we propose a method for updating the GP models when improved estimates become available. Leveraging the differentiability of the kernel function, a second-order correction algorithm is developed using the precomputed Jacobians and Hessians of the GP mean and covariance functions for real-time refinement based on measurement location discrepancy data. Simulation results demonstrate improved prediction accuracy and computational efficiency compared to full model retraining.

💡 Research Summary

The paper addresses a practical problem in robotic field mapping: the degradation of Gaussian Process (GP) regression performance when the robot’s measured locations are inaccurate. While standard GP regression assumes perfectly known input coordinates, real‑world mobile robots rely on GPS, SLAM, or dead‑reckoning, which introduce deterministic, often non‑stationary errors such as drift. Existing approaches either treat input uncertainty as a random Gaussian variable (e.g., Girard’s analytical propagation for the squared‑exponential kernel) or use a first‑order Taylor expansion (NIGP) to approximate the effect of noisy inputs. These methods suffer from limited kernel applicability, loss of accuracy for highly nonlinear fields, and additional hyper‑parameter optimization overhead.

The authors propose a deterministic correction framework that updates an already‑trained GP model when improved location estimates become available, without retraining the full model. The key idea is to perform a second‑order Taylor series expansion of the GP posterior mean and covariance with respect to the training inputs. Let the planned measurement locations be (\hat X) and the true locations be (X = \hat X + \delta), where (\delta_i) is the known localization error for the i‑th point. The corrected mean (M) and covariance (S) are expressed as

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